# Sobolev W0 basics question

1. Oct 5, 2008

### maze

As I understand it, one way to define Sobolev spaces is to say they are the collection of functions with weak derivatives up to some order, and this space is called $W^{k,p}$.

On the other hand, some stuff I am reading now defines it differently. Here they define the Sobolev space $W_0^{k,p}$ as the completion of $C_0^\infty$ (the space of smooth functions with compact support) with respect to the Sobolev norm.

Now, by the definition elements of $W_0^{k,p}$ are equivalence classes of Cauchy sequences in $C_0^\infty$. How do we know that these equivalence classes actually represent functions? For example, what does it even mean to "integrate" an equivalence class?